Mathematics

The key insight is that if a chord is bisected by the line $y = x$, then the midpoint of the chord lies on $y = x$, and the chord itself must be perpendicular to the line joining the center to that midpoint. Start by identifying the circle's center and radius from the given equation, then use the fact that for a chord bisected at point $(t, t)$ on the line $y = x$, the line from the center to $(t, t)$ is perpendicular to the chord. Since the chord passes through $(a, b+c)$, you can set up a geometric condition: the vector from the center to the midpoint must be perpendicular to the vector from the midpoint to the point $(a, b+c)$.

The key insight is that if a chord is bisected by the line $y = x$, then the midpoint of the chord lies on $y = x$, and the chord itself must be perpendicular to the line joining the center to that midpoint. Start by identifying the circle's center and radius from the given equation, then use the fact that for a chord bisected at point $(t, t)$ on the line $y = x$, the line from the center to $(t, t)$ is perpendicular to the chord. Since the chord passes through $(a, b+c)$, you can set up a geometric condition: the vector from the center to the midpoint must be perpendicular to the vector from the midpoint to the point $(a, b+c)$.